Then there's your problem: the OpenSSL 1.1.0 release broke API and ABI compatibility with the 1.0 release. There's currently a new backend being written for that new version but in between you have to install the libssl1.0-dev library and compile against that one.

@SGaist
Does Qt have the official public bugtracker or smthing similar, where each Qt release brief desrciption comes with a list of dependencies ( where versions of these dependencies are pointed precisely)? One wants to see all possible issues and caveats.

Yes, there's an official bug tracker that's mentioned numerous of time on the forum as well as on the official Qt website and it's quite easy to find. A quick search with "qt openssl 1.1" would have pointed you the related bug reports.

On the good news front, following QTBUG-52905, Qt 5.10 should have support for OpenSSL 1.1.

As for the release informations, you have all the change notes in the qtname_of_module/dist folder. The Qt wiki also provides pages for each release.

@SGaist
It turns out that I can't downgrade the current version of libssl-dev with the package management tool because 1.1.0 - minimal available version in repos. Ok, I tried to launch configure with -no-ssl
(./configure -static -release -opensource -confirm-license -qt-zlib -qt-pcre -qt-libpng -qt-libjpeg -qt-freetype -no-ssl -opengl desktop -prefix /home/tikani/Qt5.9.1/5.9.1/static)
followed by make, but ...

Like I wrote before Debian provides libssl1.0-dev in stretch for that kind of case.

When you modify the configure parameter you should first cleanup. In fact, you should rather do out of tree builds so if something goes wrong you can simply nuke the build folder or if you want to build with several different parameters set, you can just change to a new folder.